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Algebra 2 Interactive Chalkboard
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GLENCOE DIVISION
Glencoe/McGraw-Hill
8787 Orion Place
Columbus, Ohio 43240
Lesson 13-1 Right Triangle Trigonometry
Lesson 13-2 Angles and Angle Measure
Lesson 13-3 Trigonometric Functions of General Angles
Lesson 13-4 Law of Sines
Lesson 13-5 Law of Cosines
Lesson 13-6 Circular Functions
Lesson 13-7 Inverse Trigonometric Functions
Example 1 Find Trigonometric Values
Example 2 Use One Trigonometric Ratio to Find Another
Example 3 Find a Missing Side Length of a
Right Triangle
Example 4 Solve a Right Triangle
Example 5 Find Missing Angle Measures of Right
Triangles
Example 6 Indirect Measurement
Example 7 Use an Angle of Elevation
Find the value of the six
trigonometric functions
for angle G.
For this triangle, the leg opposite
adjacent to
The hypotenuse is
and the leg
Use opp = 24, adj = 32, and hyp = 40 to write each
trigonometric ratio.
Answer:
Find the value of the six trigonometric
functions for angle A.
Answer:
Multiple-Choice Test Item
If
find the value of csc A.
A
B
C
D
Read the Test Item
Draw a right triangle and label
one acute angle A. Since
and
, label the opposite
leg 5 and the adjacent leg 3.
Solve the Test Item
Use the Pythagorean Theorem to find c.
Pythagorean Theorem
Replace a with 3 and b with 5.
Simplify.
Take the square root
of each side.
Now find csc A.
Cosecant ratio
Replace hyp with
and opp with 5.
Answer: D
Multiple-Choice Test Item
If
find the value of cos B.
A
B
C
D
Answer: C
Write an equation involving sin, cos,
or tan that can be used to find the
value of x. Then solve the equation.
Round to the nearest tenth.
The measure of the hypotenuse is 12.
The side with the missing length is
opposite the angle measuring 60. The
trigonometric function relating the
opposite side of a right triangle and the
hypotenuse is the sine function.
Sine ratio
Replace  with 60°, opp with x,
and hyp with 12.
Multiply each side by 12.
Answer: The value of x is
or about 10.4.
Write an equation involving sin, cos,
or tan that can be used to find the
value of x. Then solve the equation.
Round to the nearest tenth.
Answer:
or about 8.7
Solve XYZ. Round measures
of sides to the nearest tenth
and measures of angles to the
nearest degree.
x
z
You know the measures of one side, one acute angle,
and the right angle. You need to find x, z, and Y.
Find x and z.
Multiply each side by 11.
Use a calculator.
x
Multiply each side by 11.
Use a calculator.
z
Find Y.
Angles X and Y are
complementary.
Solve for Y.
Answer: Therefore,
,
, and
.
Solve XYZ. Round measures of sides
to the nearest tenth and measures of
angles to the nearest degree.
Answer:
Solve ABC. Round measures
of sides to the nearest tenth
and measures of angles to the
nearest degree.
You know the measures of the sides. You need to
find A and B.
Find A.
Use a calculator and the SIN–1 function to find the angle
whose sine is
Keystrokes: 2nd [SIN–1] 9
17 )
ENTER
31.96571875
Find B.
Angles A and B are
complementary.
Solve for B.
Answer: Therefore,
and
.
Solve ABC. Round measures
of sides to the nearest tenth
and measures of angles to the
nearest degree.
Answer:
Bridge Construction In order to
construct a bridge across a river,
the width of the river must be
determined. A stake is planted on
one side of the river directly across
from a second stake on the
opposite side. At a distance 30
meters to the left of the stake, an
angle of 55 is measured between
the two stakes. Find the width of
the river.
Let w represent the width of the river. Write an
equation using a trigonometric function that involves
the ratio of w and 30.
Multiply each side by 30.
Use a calculator.
Answer: The width is about 42.8 meters.
Bridge Construction To construct a bridge, a stake
is planted on one side of the river directly across
from a second stake on the opposite side. The
angle between the two stakes is measured at a
distance 20 meters away from the stake, and found
to be 50. Find the width of the river.
Answer: about 23.8 meters
Skiing A run has an angle of elevation of 15.7 and
a vertical drop of 1800 feet. Estimate the length of
this run.
Let represent the length of the run. Write an
equation using a trigonometric function that
involves the ratio of
and 1800.
Solve for
Use a calculator.
Answer: The length of the run is about 6652 feet.
Skiing A run has an angle of elevation of 23 and a
vertical drop of 1000 feet. Estimate the length of
this run.
Answer: about 2559 feet
Example 1 Draw an Angle in Standard Position
Example 2 Convert Between Degree and
Radian Measure
Example 3 Measure an Angle in Degrees and Radians
Example 4 Find Coterminal Angles
Draw the angle 210 in standard position.
Draw the terminal side of the angle 30 counterclockwise
past the negative x-axis.
Answer:
Draw the angle –45 in standard position.
The angle is negative. Draw the terminal side 45
clockwise from the positive x-axis.
Answer:
Draw the angle 540 in standard position.
Draw the terminal side 180 counterclockwise past the
positive x-axis.
Answer:
Draw the angle with the given measure in
standard position.
a. 225
Answer:
b. –60
c. 480
Answer:
Answer:
Rewrite 30 in radians.
Answer:
Rewrite
in degrees.
Answer: –300
Rewrite the degree measure in radians and the
radian measure in degrees.
a. 45
Answer:
b.
Answer: 30
Find both the degree and radian measures of the
angle through which the hour hand on a clock
rotates from 6 P.M. to 7 P.M.
The numbers on a clock divide it
into 12 equal parts with 12 equal
angles. The angle from 6 to 7 on
a clock represents
of a
complete rotation of 360.
of 360 is 30.
Since the rotation is clockwise, the angle through
which the hour hand rotates is negative.
Answer: Therefore, the angle measures –30. The
equivalent radian measure is
Find both the degree and radian measures of the
angle through which the hour hand on a clock
rotates from 5 P.M. to 8 P.M.
Answer: –90 or
Find one angle with positive measure and one
angle with negative measure coterminal with 210.
Sample answer:
A positive angle is
A negative angle is
Find one angle with positive measure and one
angle with negative measure coterminal with
Sample answer:
A positive angle is
A negative angle is
.
Find one angle with positive measure and one angle
with negative measure coterminal with each angle.
a. 150
Sample answer: 510, –210
b.
Sample answer:
Example 1 Evaluate Trigonometric Functions for a
Given Point
Example 2 Quadrantal Angles
Example 3 Find the Reference Angle for a Given Angle
Example 4 Use a Reference Angle to Find a
Trigonometric Value
Example 5 Quadrant and One Trigonometric Value of 
Example 6 Find Coordinates Given a Radius and
an Angle
Find the exact values of the
six trigonometric functions
of  if the terminal side of 
contains the point (8, –15).
From the coordinates given, you know
that x = 8 and y = –15.
Use the Pythagorean Theorem to find r.
Pythagorean Theorem
Replace x with 8 and y with –15.
Simplify.
Now use x = 8, y = –15, and r = 17 to write the ratios.
Answer:
Answer:
Find the exact values of the six trigonometric
functions of  if the terminal side of  contains
the point (–3, 4).
Answer:
Find the values of the six
trigonometric functions for
an angle in standard position
that measures 180.
When  = 180, x = –r, and y = 0.
Answer:
Answer:
Find the values of the six trigonometric functions for
an angle in standard position that measures 90.
Answer:
Sketch 330. Then find its reference angle.
Answer: Because the terminal side of
330 lies in quadrant IV, the reference
angle is 360 – 330 or 30.
Sketch
. Then find
its reference angle.
Answer: A coterminal angle of
Because the terminal side of this angle
lies in Quadrant III, the reference angle
is
a. Sketch 315. Then find its reference angle.
Answer: 45
b. Sketch
Answer:
. Then find its reference angle.
Find the exact value of
.
Because the terminal side of 135 lies in Quadrant II,
the reference angle
Answer: The sine function is positive in Quadrant II,
so,
Find the exact value of
Because the terminal side of
lies in Quadrant I,
the reference angle is
The cotangent function is positive in Quadrant I.
Answer:
Find the exact value of each trigonometric function.
a.
Answer:
b.
Answer:
Suppose
is an angle in standard position whose
terminal side is in Quadrant III and
Find
the exact values of the remaining five trigonometric
functions of
.
Draw a diagram of this angle labeling a point P(x, y)
on the terminal side of
Use the definition of
cosecant to find the values of y and r.
Given
Definition
of cosecant
Since y is a negative in Quadrant III and r is always
positive,
and
Use these values and the
Pythagorean Theorem to find x.
Pythagorean Theorem
Replace y with –3 and r with 5.
Square the numbers.
Subtract 9 from each side.
Take the square root of each side.
x is negative in Quadrant III.
to write the remaining
trigonometric ratios.
Answer:
Suppose
is an angle in standard position whose
terminal side is in Quadrant II and
Find
the exact values of the remaining five trigonometric
functions of
Answer:
.
Robotics In a robotics competition, a robotic arm is
used to pick up an object at point A and release it at
point B. Find the new position of the object relative to
the pivot point O for a robotic arm that is 3 meters long
and that rotates through an angle of 150.
With the pivot point at the origin and the angle through
which the arm rotates in standard position, point A has
coordinates (0, 3). The reference angle
for 150 is
180 – 150 or 30.
Let the position of point B have coordinates (x, y). Then,
use the definitions of sine and cosine to find the value of x
and y. The value of r is the length of the robotic arm, 3
meters. Because B is in Quadrant II, sine of 150 is
positive and cosine of 150 is negative.
Cosine ratio
Solve for x.
Sine ratio
Solve for y.
Answer: The exact coordinates of B are
Since
is about 2.60, the object is about 2.60
meters to the left of the pivot point and 1.5 meters in
front of the pivot point.
Robotics In a robotics competition, a robotic arm is
used to pick up an object at point A and release it at
point B. Find the new position of the object relative to
the pivot point O for a robotic arm that is 2 meters long
and that rotates through an angle of 120.
Answer:
Example 1 Find the Area of a Triangle
Example 2 Solve a Triangle Given Two Angles and
a Side
Example 3 One Solution
Example 4 No Solution
Example 5 Two Solutions
Example 6 Use the Law of Sines to Solve a Problem
Find the area of
to the nearest tenth.
In this triangle,
Use the formula
Area formula
Replace b with 6, c with 3,
and A with 25.
Use a calculator.
Answer: To the nearest tenth, the area is
3.8 square centimeters.
Find the area of
Answer:
to the nearest tenth.
Solve
.
You are given the measures of two angles and a side.
First, find the measure of the third angle.
The sum of the angle measures
of a triangle is 180.
Use the Law of Sines to find a and c.
Law of Sines
Replace A with 27, B with 53,
C with 100, and b with 9.
Solve for the variable.
Use a calculator.
Law of Sines
Replace A with 27, B with 53,
C with 100, and b with 9.
Solve for the variable.
Use a calculator.
Answer:
Solve
Answer:
.
In
Determine
whether
has no solution, one solution or two
solutions. Then solve
Angle A is acute and
so one solution exists. Make
a sketch and then use the
Law of Sines to find B.
Law of Sines
Multiply by 12.
Use a calculator.
Use the
function.
The measure of angle C is approximately
Use the Law of Sines again to find c.
Law of Sines
or about 22.9
Answer:
In
Determine
whether
has no solution, one solution or two
solutions. Then solve
Answer: one;
In
Determine
whether
has no solution, one solution or two
solutions. Then solve
Answer:
and A is obtuse, so there is no solution.
In
Determine
whether
has no solution, one solution or two
solutions. Then solve
Answer: no solution
In
Determine
whether
has no solution, one solution or two
solutions. Then solve
Since angle A is acute, find b sin A and compare it with a.
Replace b with 10 and A with 25.
Use a calculator.
Since
there are two possible solutions.
Thus, there are two possible triangles to be solved.
Case 1
Acute Angle B
First use the Law of Sines to find B.
The measure of angle C is approximately
Use the Law of Sines again to find c.
Answer: Therefore,
Case 2
Obtuse Angle B
To find B, you need to find an obtuse angle whose sin is
also 0.8452. To do this, subtract the angle given by your
calculator, 58, from 180. So B is approximately 180 – 58
or 122.
The measure of angle C is approximately
Use the Law of Sines to find c.
Answer: Therefore,
In
Determine
whether
has no solution, one solution or two
solutions. Then solve
Answer: two;
Lighthouses A lighthouse is
located on a rock at a certain
distance from a straight shore.
The light revolves counterclockwise at a steady rate of
one revolution per minute. As
the beam of light revolves, it
strikes a point on the shore
that is 1840 feet from the
lighthouse. Two seconds later,
the light strikes a point 500 feet
farther down the shore. To
the nearest foot, how far is the
lighthouse from the shore?
Because the lighthouse makes one revolution every 60
seconds, the angle through which the light revolves in
2 seconds is
Use the Law of Sines to find the measure of angle
Law of Sines
Multiply by 1840.
Use a calculator.
Use the sin–1 function.
Use this angle measure to find the measure of angle
Since
is a right triangle, the measures of angle
and
are complementary.
Angles and
are complementary.
Simplify.
Solve for
To find the distance from the lighthouse to the shore,
solve
Cosine ratio
Solve for d.
Use a calculator.
Answer: The distance from the lighthouse to the
shore, to the nearest foot, is 1625 feet.
Lighthouses A lighthouse is located on a rock at a
certain distance from a straight shore. The light
revolves counterclockwise at a steady rate of one
revolution per minute. As the beam of light revolves, it
strikes a point on the shore that is 540 feet from the
lighthouse. Two seconds later, the light strikes a point
500 feet farther down the shore. To the nearest foot,
how far is the lighthouse from the shore?
Answer: 228 feet
Example 1 Solve a Triangle Given Two Sides and
Included Angle
Example 2 Solve a Triangle Given Three Sides
Example 3 Apply the Law of Cosines
Solve
You are given the measures of two
sides and the included angle. Use the
Law of Cosines to find c.
Law of Cosines
Simplify using
a calculator.
Take the square
root of each side.
Next, use the Law of Sines to find the measure of angle A.
Law of Sines
Multiply each side by 7.
Use a calculator.
Use the sin–1 function.
The measure of angle B is approximately
Answer: Therefore,
Solve
Answer:
Solve
You are given the measures
of three sides. Use the Law
of Cosines to find the
measure of the largest
angle first, angle C.
Law of Cosines
Divide each side by –126.
Use a calculator.
Use the cos–1 function.
Use the Law of Sines to find the measure of angle B.
Law of Sines
Multiply each side by 7.
Use a calculator.
Use the sin–1 function.
The angle of measure A is approximately
Answer: Therefore,
Solve
Answer:
Emergency Medicine A helicopter flies 55 miles from
its base at point C to an accident at point B and then
35 miles from there to the hospital at point A. Angle B
equals 42. How far will the helicopter have to fly to
return to its base from the hospital?
You are given the measures of two sides and their
included angle. Use the Law of Cosines to find b.
Law of Cosines
Use a calculator
to simplify.
Take the square root of
each side.
Answer: The distance between the hospital and the
helicopter base is approximately 37.3 miles.
Emergency Medicine A helicopter flies 60 miles from
its base at point C to an accident at point B and then
25 miles from there to the hospital at point A. Angle B
equals 55. How far will the helicopter have to fly to
return to its base from the hospital?
Answer: about 50.0 miles
Example 1 Find Sine and Cosine Given Point on
Unit Circle
Example 2 Find the Value of a Trigonometric Function
Example 3 Find the Value of a Trigonometric Function
Given an angle
in standard position, if
lies on the terminal side of
find sin
Answer:
and cos .
and on the unit circle,
Given an angle
in standard position, if
lies on the terminal side of
find sin
Answer:
and cos .
and on the unit circle,
Find the exact value of
Answer:
.
Find the exact value of
Answer:
.
Find the exact value of each function.
a.
Answer:
b.
Answer: –1
Ferris Wheel As you ride a Ferris wheel, the height that
you are above the ground varies periodically as a
function of time. Consider the height of the center of
the wheel to be the starting point. A particular Ferris
wheel has a diameter of 42 feet, and travels at a rate of
3 revolutions per minute.
Identify the period of this function.
Since the wheel makes 3 complete revolutions per minute,
the period is the time that it takes to complete one
revolution, which is
Answer: 20 seconds
of a minute or 20 seconds.
Make a graph in which the horizontal axis represents
the time in t seconds and the vertical axis represents
the height h in feet in relation to the starting point.
The diameter of the wheel is 42 feet, so the wheel reaches
a maximum height of
or 21 feet above the starting
point and a minimum of 21 feet below the starting point.
Answer:
Ferris Wheel On another Ferris wheel, the diameter of
the wheel is 30 feet, and it travels at a rate of 5
revolutions per minute.
a. Identify the period of this function.
Answer: 12 seconds
b. Make a graph in which the horizontal axis represents
the time in t seconds and the vertical axis represents
the height h in feet in relation to the starting point.
Answer:
Example 1 Solve an Equation
Example 2 Apply an Inverse to Solve a Problem
Example 3 Find a Trigonometric Value
by finding the value of x to the
Solve
nearest degree.
If
So,
, then x is the least value whose sine is
Use a calculator to find x.
Keystrokes:
2nd [SIN–1] 2nd
45
Answer: Therefore,
2
)
2
)
Enter
Solve
nearest degree.
Answer:
by finding the value of x to the
Drawbridge Each leaf of a certain double-leaf
drawbridge is 130 feet long. What is the minimum angle
to the nearest degree, to
which each leaf should
open so that a ship that
is 100 feet wide will fit?
When two parts of the bridge are in their lowered position,
the bridge spans
In order for the
ship to fit, the distance between the leaves must be at
least 100 feet.
This leaves a horizontal distance of
or 80 feet
from the pivot point of each leaf to the ship as shown in
the diagram.
To find the measure of angle
right triangles.
use the cosine ratio for
Cosine ratio
Replace adj with 80,
hyp with 130.
Inverse cosine function
Use a calculator.
Answer: Thus, the minimum angle through which
each leaf of the bridge should open is 52.
Drawbridge Each leaf of a certain double-leaf
drawbridge is 130 feet long. What is the minimum angle
to the nearest degree, to which each leaf should
open so that a ship that is 150 feet wide will fit?
Answer: 65
Find the value of Arcsin
Write the angle measure
in radians. Round to the nearest hundredth.
Keystrokes:
2nd [SIN–1] 2nd
.7853981634
Answer:
2
)
2
)
Enter
Find the value of
Write the angle
measure in radians. Round to the nearest hundredth.
Keystrokes:
TAN
Answer:
2nd
[COS–1]
4
5
)
Enter
0.75
Find each value. Write the angle measures in radians.
Round to the nearest hundredth.
a.
Answer: 1.05 radians
b.
Answer: 1.12 radians
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